Download Tutorial Questions 4

Tutorial Questions 4
ALGEBRAIC EQUATIONS 2
Simultaneous Linear Equations: Two unknowns
1.
Solve the following simultaneous equations:
(a)
6x – 5y = -7
(b)
4x + 2y + 8 = 0
3x + 4y = 16
6x = 2y – 27
(c)
(e)
4 x  5 y  335
7 x  4 y  37
9 x  14 y  850
(d)
5 x  3 y  51
4x – 3y = 5
6x + y =2
(f)
4x + 2y + 11 = 0
3x + 4y + 5 = 0
(g)
3x + 7y = 13
(h)
3x + 4y = 14
2x = 4 + 7y
-2x + y= 9
[Answer: (a) x = 4/3, y = 3 (b) x = - 3.5, y = 3, (c) x = 40, y = 35,
(d) x = -93, y = -172 (e) x = ½, y = -1 (g) x = 3.4, y = 0.4 (h) x =-2, y=5]
2.
Solve the following simultaneous equation:
(a)
x + 4y = 3,
(b)
3x – 4y = 13,
3x – 2y = -5
2x + 3y = 3.
(c)
5v + 2w = 36
(d)
x – 2y = 8,
8v – 3w = -54
5x + 3y = 1
[Answer: (a) x = -1, y = 1, (b) x = 3, y= -1, (c) v = 0, w = 18, (d) x = 2, y = - 3]
3.
Solve the simultaneous equations:
(a)
y = 3x + 5;
(b)
x + 3y = 7
(c)
2y + 3x = 28
2x – 2y = 6
[Answer: (a) x = 2, y = 11 (b) x = 4, y = 1 (c) x = 7, y = 3]
4.
7x – 4y = 37
6x + 3y =51
Use the method of elimination to find the value of x and y that satisfy the following
simultaneous equations:
(a)
x+y=4
(b)
3x + 2y = 19
(c)
3x + 2y =7
3x – y = 8
3x – 3y = -6
x + y =3
(d)
3x  4 y  5
(e)
7 x  3 y  41
(f)
4 x  3 y  18.5
x  7 y  10
3x  y  17
4 y  7 x  35.5
[Answer: (a) x = 3, y = 1, (b) x = 3, y = 5, (c) x = 1, y = 2 (d) x = 5, y = -2
(e) x = 5, y = - 2 (f) x = 6.5, y = 2.5]
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5.
Solve the following simultaneous equations:
(a)
(c)
2
3
x  12 y  2
3
8
x  65 y   112
4x
3
x
2
 32y  4

y
4
(b)
(d)
 1
x  2 y  1 13


3
4
12
2  x 3  y 11


2
3
6
3 4x  y 
4
4
3 2x  3y 
 2x  0
5
x 1
2
y 1
(e)
2x  1 1

2y 1 3
[Ans.: (a) x = 12, y = - 12, (b) x = 3, y= 4, (c) x = - 6, y =8, (d) x = 3/2, y = - ⅔ (e) x = 3/5, y = -4/5]
6.
Solve for the unknowns in the following sets of equations:
7
5
12( x  y )  11 y  14
4 x 3 y3 0
(a)
(b)
y
x
18 x  5( y  1)  0
2  3  4
(d)
(f)
(h)
x 1
2

2 x 1
3

y 3
4
x  y  8000
0.04 x  0.06 y  0.0525( x  y )
x y4
4
3 x y 2
3
(j)
[Answer:


2 x y
3
2 x y
4
p
(e)
(g)
(i)
x y2
7
 3x 
y
2
0.5 x  0.3 y  1.65
0.7 x  0.2 y  0.14
y  2 z 1
4

3 y  z 1
8

2 y 3 z  2
9
3  9(27) q
log 2 7  log 2 (11q  2 p)  1
(l) 6.5, 1.5]
7.
Solve the following simultaneous equation:
3 4
 2
5
8
1
1
x y
x  y 3
x  y  12
(a)
(b)
(c)
3
3
2
2
4 1
x  y  7
x  y  14
 3
x y
[Ans.: (a) x = 19/14, y = -19 (b) x = ½, y = 1 (c) x = ¼, y = 1]
9.
A bill for $74 was paid with $5 and $1 notes, a total of 50 notes being used. How many
$5 notes were there?
[Answer: 6]
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10.
A company manufactures two products X and Y by means of two processes A and B. The
maximum capacity of process A is 1750 hours and of process B 4000 hours. Each unit of
product X requires 3 hours in A and 2 hours in B, while each unit of product Y requires 1
hour in A and 4 in B. Use the algebraic method to calculate how many units of products
X and Y are produced if the maximum capacity available is utilized.
[Answer: 300 units of x and 85 units of y]
11.
Two numbers are such that one third of the smaller number plus half the larger number is
19. If the difference between the two numbers is 13, find the two numbers.
[Answer: x = 28, y = 15]
12.
500 tickets were sold for a concert, some at 40¢ each and the remainder at 25¢ each. The
money received for the dearer tickets was $70.00 more than for the cheaper tickets. Find
the number of dearer tickets sold.
[Answer: 300]
13.
300 tickets were sold for a concert, some at $8 each and the remainder at $5 each. The
money received for the dearer tickets was $60 more than for the cheaper tickets. Find the
number of dearer tickets which were sold.
[Answer: 120]
14.
76 is divided into two parts, x and y respectively, so that ⅔ of the first part is equal to
of the second part. Find the two parts x and y.
[Answer: x = 36, y = 40]
15.
A chemical manufacturer wishes to fill an order for 500 liters of a 25% acid solution.
(Twenty-five percent by volume is acid.) If solutions of 30% and 18% are available in
stock, how many liters of each must be mixed to fill the order?
[Answer: 291⅔ liters of 30%, 208⅓ liters of 18%]
16.
A box contains 44 coins, made up of dimes (10c) and nickels (5c coins). If the total value
of the coins is $3.50, calculate the number of dimes in the box.
[Answer: 26]
17.
A man went to a post office to buy some stamps. If he bought x, 50 cents stamps and y,
25 cents stamps, the total cost would have been $3.50. If however he bought twice as
many 50 cent stamps and half as many 25 cents stamps, then the total cost would have
been $4.75. Find x and y.
[Answer: x = 4, y = 6]
18.
An invoice clerk receives a bill for £37.50 for ten blank ledger books and three special
filing trays. On phoning the supplier (after discovering the order had been written out
incorrectly) the clerk agrees to return three of the ledger books in return for the supplier
sending an extra filing tray. Given that there will be an extra £2.50 to pay and that there is
a 10% discount for orders of 10 ledger books or more:
(a)
derive linear equations in x and y to represent the components of the
original and revised invoices, where x is the price of a single ledger book and y
is the price of a filing tray, and
(b)
solve the two equation simultaneously for x and y.
[Answer: (a) 10(0.9)x + 3y = 37.5, 7x + 4y = 40, (b) x = 2, y = 6.5]
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19.
A furniture factory manufactures two type of coffee table, A and B. each table goes
through two distinct costing stages, assembly and finishing. The maximum capacity for
assembly is 195 hours and for finishing, 165 hours. Each A table requires 4 hours
assembly and 3 hours finishing, while a B table requires 1 hour for assembly and 2 hours
for finishing. Calculate the number of A and B tables to be produced to ensure that the
maximum capacity available is utilized.
[Answer: 45A and 15B tables]
20.
A chemical manufacturer wishes to fill an order for 700 gallon of a 24% acid solution.
Solutions of 20% and 30% are in stock. How many gallons of each solution must be
mixed to fill the order?
[Answer: 420 (20%), 280 (30%)]
21.
A manufacturer of dining-room sets produces two styles: early American and
contemporary. From past experience, management has determined that 20% more of the
early American styles can be sold than the contemporary styles. A profit of $250 is made
on each early American set sold, whereas a profit of $350 is made on each contemporary
set. If, in the forthcoming year, management desires a total profit of $130.000, how many
units of each style must be sold?
[Answer: 240(EA), 200]
22
When petrol cost 10¢ a litre and oil 25¢ a litre, a motorist found that his cost in petrol and
oil was $10.50 for each 1000km. When petrol was increased to 10½¢ a litre and oil to
26¢ a litre, the cost was $11.02. Find how many litres of petrol and oil he used to travel
1000km.
[Answer: 100 litres of petrol; 2 litres of oil]
23.
A sum of $2800 is invested, partly at 5% and partly at 4%. If the total income is $128 per
annum, find the amount invested at 5%.
[Answer: $1600]
24.
A supervisor and 7 workers together earn $520 per week, whilst 2 supervisors and 17
workers together earn $1220 per week. Find the weekly wage of a supervisor and of a
worker.
[Answer: $100, $60]
25.
In the year 300 B.C., the Mathematician Euclid gave the following problem to his
student: ‘A mule and a donkey were going to a market laden with wheat. The mule said:
“If you give me one measure, I should carry twice as much as you, but if I give you one
measure we shall have equal burden.”’ Calculate the burden of the mule and that of the
donkey.
[Answer: 7; 5]
26.
A bird laid a certain number of eggs in a nest, and a collector took away two-thirds of
them. The bird laid some more, and another collector took two-thirds of them. If the bird
laid 12 eggs altogether and collectors took 10 altogether, how many eggs were there to
begin with?
[Answer: 9]
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27.
A sum of $3000 is invested, partly at 3½% and partly at 4%. If the annual income from
the two investments $112, find how much is invested at 4%.
[Answer: $1400]
28.
The charge for electricity is 3¢ per unit for lighting and ½¢ per unit for heating. A man’s
bill for a quarter should have been $8. By mistake, his lighting was charged at ½¢ per
unit and his heating at 3¢ per unit. The amount of the bill was $13. How many units of
electricity did he use altogether?
[Answer: 600]
29.
At a concert, tickets were 70¢ or 50¢ each. Three quarters of those who paid 70¢ bought
programmes and half the 50¢ seat-holders bought programmes. The receipts from the
tickets were $580 and from the programme $60. If the programmes cost 10¢ each, how
many paid 70¢ for the concert?
[Answer: 400]
30.
The cost $C of a journey on a certain railway is given by the formula: C = a + bx, where
‘x’ is the number of kilometers traveled and ‘a’ and ‘b’ are constants. A journey of 50km
costs $2, and a journey of 70km costs $2.50. Find the cost of a journey of 90km.
[Answer: $3]
31.
If I invest $2000 at x% and $2500 at y%, my annual income is $160. Had I invested
$2500 at x% and $2000 at y%, my income would have been $155. Find x and y.
[Answer: 3; 4]
32.
I invest $x at 4% and $y at 5%. My annual income is $230. Had I invested $x at 5% and
$y at 4%, my annual income would have been $220. Find x and y. [Answer: 2000; 3000]
33.
At a meeting of a cricket club to elect a captain, 75 members were present, and the
captain was elected by a majority of 13, all voting. How many voted for and against?
[Answer: 44 for; 31 against]
34.
The daily wages of 10 men and 7 boys amount to $40.32; if a man earns in two days as
much as a boy earns in seven days, find what each earns per day.
[Answer: $3.36; $0.96]
35.
If Ada were to give Ellis twelve dollars, Ada would have half the sum which Ellis then
has; but, if Ellis were to give Ada thirteen dollars, Ellis would have one-third of what
Ada has. How much money has each originally?
[Answer: $32; $28]
36.
A number of two digits is such that the sum of its digits is 10. When the digits are
reversed, the number is increased by 36. Find the number.
[Answer: 37]
37.
A number of two digits is such that twice the ten digit is 6 greater than the unit digit.
When the digits are reversed, the number is increased by 9. What is the number?
[Answer: 78]
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38.
A number with three digits has the hundred digit three times the unit digit and the sum of
the digits is 19. If the digits are written in reverse order, the value of the number is
decreased by 594. Find the number.
[Answer: 973]
39.
Solve the simultaneous equations:
125x
and
 625
25 y
[Answer: x = 2, y = 1]
53 x
 3125 and
25 y
find the value of x and of y.
[Answer: x = 2, y = 2.5]
(2)(4 x )  32 y
2 x 4( y1)  32 ,
40.
Given that:
41.
5x 252 y  1
35 x 9 y  19 ,
If
and
calculate the value of x and of y.
[Answer:
x  94 , y  19 ]
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